Comparisons of Relative BV-Capacities and Sobolev Capacity in Metric Spaces

Comparisons of relative BV-capacities and Sobolev capacity in metric spaces

H. Hakkarainen∗

Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland.

N. Shanmugalingam Department of Mathematical Sciences, P.O.Box 210025, University of Cincinnati, Cincinnati, OH 45221–0025, U.S.A.

Abstract We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak (1, 1)-Poincar´einequality. We prove the equality of 1-modulus and 1-capacity, extending the known results for 1 < p < ∞ to also cover the more geometric case p = 1. Then we give alternative deﬁnitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity. Keywords: Capacity, Functions of Bounded Variation, Analysis on Metric Measure Spaces 2010 MSC: 28A12, 26A45, 30L99

1. Introduction

In this article we study connections between different 1-capacities and BV-capacities in the setting of metric measure spaces. We obtain characterizations for the 1-capacity of a condenser, extending results known for p > 1 to also cover the more geometric case p = 1. Furthermore, we study how the different versions of BV-capacity, corresponding to various pointwise requirements that the capacity test functions need to fulfill, relate to each other. One difficulty that occurs when working with minimization problems on the space W 1,1(Rn) is the lack of reflexivity. Indeed, the methods used to develop the theory of p-capacity are closely related to those used in certain variational minimization problems; in such problems reflexivity or the weak compactness property of the function space W 1,p(Rn) when p > 1 usually plays an important role. One possible way to deal with

∗Corresponding author Email addresses: [email protected] (H. Hakkarainen), [email protected] (N. Shanmugalingam) Preprint submitted to Elsevier 28th March 2011 this lack of reflexivity is to consider the space BV(Rn), that is, the class of functions of bounded variation. This wider class of functions provides tools, such as lower semiconti- nuity of the total variation measure, that can be used to overcome the problems caused by the lack of reflexivity in the arguments. This approach was originally used to study variational 1-capacity in the Euclidean case in [1], [2], and [3]. The article [4] showed that similar approach could also be used to study variational 1-capacity in the setting of metric measure spaces. In [4] the main tool in obtaining a connection between the 1-capacity and BV-capacity was the metric space version of Gustin’s boxing inequality, see [5]. Since then, this strategy has been used in [6] to study a version of BV-capacity and Sobolev 1-capacity in the setting of metric measure spaces. Since the case p = 1 corresponds to geometric objects in the metric measure space such as sets of finite perimeter and minimal surfaces, it behooves us to understand this case. A study of potential theory in the case p = 1 for the setting of metric measure spaces was begun in the papers [6], [4], and [7], and we continue the study in this note. In the theory of calculus of variations, partial differential equations and potential theory many interesting features of Sobolev functions are measured in terms of capacity. Just as sets of measure zero are associated with the Lp-theory, sets of p-capacity zero should be understood in order to deal with the Sobolev space W 1,p(Rn). For 1 < p < ∞ the p-capacity of sets is quite well-understood; see for example [8], [9], [10] for the Euclidean setting, and in the more general metric measure space setting, [11], [12], [13], [14], [15], [16], [17], and the references therein. The situation corresponding to the case p = 1 is not so well-understood. In the setting of p = 1 there are two natural function spaces to consider: Sobolev type spaces, and spaces of functions of bounded variation. Since these two spaces are interrelated, it is natural to expect that the corresponding capacities are related as well. However, these two function spaces are fundamentally different in nature. Functions in the Sobolev class W 1,1(Rn) have quasicontinuous representatives, whereas functions of bounded variation need not have quasicontinuous representatives, and in fact sometimes exhibit jumps across sets of nonzero codimension one Hausdorff measure; see for example [18]. Because of the quasicontinuity of Sobolev type functions, one can either insist on the test functions to have value one in a neighborhood of the set whose capacity is being computed, or merely require the test functions to have value 1 on the set. Such flexibility is not available in computing the BV-capacities and hence different point-wise requirements on the test functions might lead to different types of BV-capacity. Thus, corresponding to the BV-class there is more than one possible notion of capacity and it is nontrivial to even know which versions of BV-capacities are equivalent. We point out here that the analog of Sobolev spaces considered in this paper, called the Newton-Sobolev spaces, consist automatically only of quasicontinuous functions when the measure on X is doubling and the space supports a (1, 1)-Poincaréinequality. The primary goal of this note is to compare these different notions of capacity related to the BV-class, and to the capacity related to the Newton-Sobolev space N 1,1(X). In Section 2 we describe the objects considered in this paper, and then in Section 3 we show that the 1-modulus of the family of curves in a domain Ω in the metric space, connecting two nonempty pairwise disjoint compact sets E,F ⊂ Ω, is the same as the Newton–Sobolev 1-capacity and local Lipschitz capacity of the condenser (E,F, Ω). Our arguments are based on constructing appropriate test functions. In Section 4 we consider 2 three alternative notions of variational BV-capacity of a compact set K ⊂ Ω, and show that these notions are comparable to each other and to the variational Sobolev 1-capacity of K relative to Ω. We combine tools such as discrete convolution and boxing inequality to obtain the desired results. In Section 5 we consider total capacities where we minimize the norm of the test functions rather than their energy seminorm. We consider these quantities of more general bounded sets - related to both BV-functions and to Sobolev functions - and compare them to their variational capacities. In the classical Euclidean setting some of these results are known, for example from [2], but even in the weighted Euclidean setting and the Carnot groups the results of this paper are new.

2. Preliminaries

In this article X = (X, d, µ) is a complete metric measure space. We assume that µ is a Borel regular outer measure that is doubling, i.e. there is a constant CD, called the doubling constant of µ, such that

µ(2B) ≤ CDµ(B) for all balls B of X. Furthermore, it is assumed that 0 < µ(B) < ∞ for all balls B ⊂ X. These assumptions imply that the metric space X is proper, that is, closed and bounded sets are compact. In this article, by a path we mean a rectiﬁable nonconstant continuous mapping from a compact interval to X. The standard tool that is used to measure path families is the following.

Definition 2.1. Let Ω ⊂ X be an open connected set. The p-modulus of a collection of paths Γ in Ω for 1 ≤ p < ∞ is defined as Z p Modp(Γ) = inf % dµ % Ω where the infimum is taken over all nonnegative Borel-measurable functions % such that R γ % ds ≥ 1 for each γ ∈ Γ. We use the definition of Sobolev spaces on metric measure space X based on the notion of p-weak upper gradients, see [11], [19]. Note that instead of the whole space X, we can consider its open subsets in the definitions below. Definition 2.2. A nonnegative Borel function g on X is an upper gradient of an extended real valued function u on X if for all paths γ, Z |u(x) − u(y)| ≤ g ds, (1) γ R whenever both u(x) and u(y) are finite, and γ g ds = ∞ otherwise. Here x and y denote end points of γ. Let 1 ≤ p < ∞. If g is a nonnegative measurable function on X, and if the integral in (1) is well defined and the inequality holds for p-almost every path, then g is a p-weak upper gradient of u.

3 The phrase that inequality (1) holds for p-almost every path with 1 ≤ p < ∞ means that it fails only for a path family with zero p-modulus. Many usual rules of calculus are valid for upper gradients as well. For a good overall reference interested readers may see [20], [21], and [22]. p If u has a p-weak upper gradient g ∈ Lloc(X), then there is a minimal p-weak upper gradient gu such that gu ≤ g µ-almost everywhere for every p-weak upper gradient g of u, see [21] and [22]. When Ω is an open subset of Euclidean space equipped with the Euclidean metric and Lebesgue measure, the Sobolev type space considered below coincides with the space of quasicontinuous representatives of classical Sobolev functions. In the setting of weighted Euclidean spaces with p-admissible weights, or the Carnot-Carathéodory spaces, the corresponding Sobolev spaces coincide in an analogous manner with the Sobolev type space considered below. We refer the interested reader to [19, Example 3.10]; recall that in the Carnot-Carathéodory setting, the classical Sobolev spaces consist of functions in Lp(X) whose horizontal derivatives are also in Lp(X), and so the upper gradient analog is with respect to paths with tangents in the horizontal directions. In the Carnot- Carathéodory setting paths with finite lengths are precisely paths with tangents in the horizontal directions. Definition 2.3. Let 1 ≤ p < ∞. If u ∈ Lp(X), let

Z Z 1/p p p kukN 1,p(X) = |u| dµ + inf g dµ , X g X where the inﬁmum is taken over all p-weak upper gradients of u. The Newtonian space on X is the quotient space

1,p N (X) = u : kukN 1,p(X) < ∞ ∼, where u ∼ v if and only if ku − vkN 1,p(X) = 0. Note that if X has no nonconstant rectifiable path, then zero is an upper gradient of every function, and so in this case N 1,p(X) = Lp(X). Therefore, to have a reason- able theory of Sobolev spaces that gives rise to a viable potential theory, one needs a further condition on the metric measure space. In literature, the following Poincarétype inequality is considered. Definition 2.4. Let 1 ≤ p < ∞. The space X supports a weak (1,p)-Poincaréinequality if there exists constants CP > 0 and τ ≥ 1 such that for all balls B(x, r) of X, all locally integrable functions u on X, and for all p-weak upper gradients g of u,

Z Z !1/p p − |u − uB(x,r)| dµ ≤ CP r − g dµ , B(x,r) B(x,τr) where Z 1 Z uB(x,r) = − u dµ = u dµ. B(x,r) µ(B(x, r)) B(x,r)

4 It is known that Lip(X) ∩ N 1,p(X) is dense in N 1,p(X) if µ is doubling and (1, p)- Poincaréinequality is satisfied, see [19]. From this it easily follows that Lipschitz functions with compact support are dense in N 1,p(X), if X is also complete. Furthermore, under a Poincaréinequality it is known that the metric space supports a myriad of rectifiable curves; see for example [11] and [23]. The theory of functions of bounded variation and sets of finite perimeter in metric measure space setting will be extensively used in this thesis. These concepts were introduced and developed in [24], [25], [26], [27],[28] and [18].

1 Deﬁnition 2.5. Let Ω ⊂ X be an open set. The total variation of a function u ∈ Lloc(Ω) is deﬁned as Z

kDuk(Ω) = inf lim inf gui dµ, {ui} i→∞ Ω where gui is an upper gradient of ui and the inﬁmum is taken over all such sequences of 1 1 functions ui ∈ Liploc(Ω) such that ui → u in Lloc(Ω). We say that a function u ∈ L (Ω) is of bounded variation, denoted u ∈ BV(Ω), if kDuk(Ω) < ∞. Furthermore, we say that 0 0 1,1 u belongs to BVloc(Ω) if u ∈ BV(Ω ) for every Ω b Ω. Note that N (Ω) ⊂ BV(Ω). It was shown in [24] that for u ∈ BVloc(X) we have kDuk(·) to be a Radon measure on X. If U ⊂ X is an open set such that u is constant on U, then kDuk(U) = 0. A Borel set E ⊂ X is said to have ﬁnite perimeter if χE ∈ BV(X); the perimeter P (E,A) of E in a Borel set A ⊂ X is the number

P (E,A) = kDχEk(A).

Remark. Since X is a complete and doubling 1-Poincar´espace, we can consider sequences 1,1 from Nloc (Ω) instead of from Liploc(Ω) in the deﬁnition of kDuk(Ω). This is due to the 1,1 fact that functions from Nloc (Ω) can be approximated by functions from Liploc(Ω). For a discussion, see for instance [29, Theorem 5.9] and [22, Theorem 5.41]. The next lemma is useful in the following sections of this paper. Lemma 2.6. Let Ω be an open subset of X and u ∈ BV(Ω) such that the support of u is a compact subset of Ω. Then there is an open set U b Ω with supt(u) b U, and a 1 sequence ui ∈ Lip(Ω) with supt(ui) ⊂ U, such that ui → u in L (Ω) and Z

kDuk(Ω) = lim gui dµ i→∞ Ω

for a choice of upper gradient gui of ui.

1 Proof. Since u ∈ BV(Ω) there is a sequence vi ∈ Liploc(Ω) with vi → u in Lloc(Ω) and Z

kDuk(Ω) = lim gvi dµ i→∞ Ω

for some choice of upper gradient gvi of vi. Since u has compact support in Ω we can ﬁnd an open set U b Ω with supt(u) b U. Let η be an L-Lipschitz function on Ω such that 0 ≤ η ≤ 1, η = 1 on supt(u), and η = 0 on Ω \ U. We set ui = η vi, and now show that this choice satisﬁes the claim of the lemma. 5 To see this, note that Z Z |u − ui| dµ = |u − η vi| dµ Ω U Z Z = |u − vi| dµ + η|vi| dµ supt(u) U\supt(u) Z Z ≤ |u − vi| dµ + |u − vi| dµ U U\supt(u) Z ≤ 2 |u − vi| dµ → 0 as i → ∞. U

Furthermore, the function gui = ηgvi + |vi| gη is an upper gradient of ui. Because gη = 0 on supt(u) ∪ (Ω \ U) and gη ≤ L, we see that Z Z Z

gui dµ ≤ gvi dµ + L |vi| dµ Ω U U\supt(u) Z Z

≤ gvi dµ + L |u − vi| dµ. Ω U\supt(u)

Because Z lim |u − vi| dµ = 0, i→∞ U\supt(u) it follows that Z

lim sup gui dµ ≤ kDuk(Ω). i→∞ Ω 1 On the other hand, by the previous argument we know that ui → u in L (Ω). Thus Z

kDuk(Ω) ≤ lim inf gui dµ, i→∞ Ω and this completes the proof. The following coarea formula holds, see [24, Proposition 4.2]: whenever u ∈ BV(X) and E ⊂ X is a Borel set, Z kDuk(E) = P ({u > t},E) dt. R It was also shown in the thesis [30, Theorem 6.2.2] that if u ∈ N 1,1(X) then Z kukN 1,1(X) = |u| dµ + kDuk(X) X and that there is a minimal weak upper gradient g of u such that whenever E ⊂ X is measurable, we have Z kDuk(E) = g dµ. E 6 The above was proved in [30, Theorem 6.2.2] for the case that E is an open set. Since kDuk(·) is a Radon measure, the above follows for all measurable sets. In a metric measure space equipped with a doubling measure, the (1, 1)-Poincar´einequality implies the following relative isoperimetric inequality, see [24] or [18]: there are constants C > 0 and λ ≥ 1 such that whenever E ⊂ X is a Borel set and B is a ball of radius r in X,

min{µ(E ∩ B), µ(B \ E)} ≤ C r P (E, λB).

See also [31, Theorem 1.1] for the converse statement. Thus if E has zero perimeter in λB, then the above inequality states that up to sets of measure zero, either B ⊂ E or B ∩ E is empty. The Hausdorﬀ measure of codimension one of E ⊂ X is deﬁned as X µ(Bi) [ H(E) = lim inf : I ⊂ N, ri ≤ δ for all i ∈ I,E ⊂ Bi . δ→0 ri i∈I i∈I

Here Bi are balls in X and ri = rad Bi. Throughout this note we will assume that (X, d, µ) is a complete metric space with a doubling measure supporting a weak (1, 1)-Poincar´einequality.

3. Modulus and Continuous capacity of condensers; the case p = 1

It was shown in [12] that when 1 < p < ∞ the p-modulus of the family of all paths in a domain Ω ⊂ X that connect two disjoint continua E,F ⊂ Ω must equal the variational continuous p-capacity of the condenser (E,F, Ω). The tools used in that paper include a strong form of Mazur’s lemma, which is not available when p = 1. On the other hand, the results from [32], which were not available at the time [12] was written, should in principle help us to overcome this difficulty. In this section we give an extension of the result of [12] to the case p = 1 by using the results of [32]. Once we have that the p-modulus of the family of curves connecting E to F in Ω equals the variational p-capacity of the condenser (E,F, Ω), we use the fact that under the Poincaréinequality Lipschitz functions form a dense subclass of the space N 1,1(X) and that functions in N 1,1(X) are quasicontinuous to prove that the capacity of the condenser can be computed by restricting to functions in N 1,1(Ω) that are continuous on Ω. An easy modification of this proof also indicates that in computing the variational Sobolev 1-capacity of a set E ⊂ Ω, one can insist that the test functions take on the value one on E, or equivalently take on the value one in a neighborhood of E, see for instance [33] and [29]. Let Ω ⊂ X be an open connected set in X, and E and F be disjoint nonempty compact subsets of Ω. Denote by Modp(E,F, Ω) the p-modulus of the collection of all rectifiable paths γ in Ω with one endpoint in E and other endpoint in F . The p-capacity of the condenser (E,F, Ω) is defined by Z p capp(E,F, Ω) = inf g dµ, Ω where the infimum is taken over all nonnegative Borel measurable functions g that are upper gradients of some function u, 0 ≤ u ≤ 1 with the property that u = 1 in E and 7 u = 0 in F . Note that in this definition it is not even required of u to be measurable. If u is in addition assumed to be locally Lipschitz, then the corresponding number obtained is denoted locLip − capp(E,F, Ω). The definitions immediately imply that for 1 ≤ p < ∞,

Modp(E,F, Ω) ≤ capp(E,F, Ω) ≤ locLip − capp(E,F, Ω).

By proposition 2.17 in [11], for 1 ≤ p < ∞ we have

Modp(E,F, Ω) = capp(E,F, Ω).

Our goal in this section is to prove that

cap1(E,F, Ω) = locLip − cap1(E,F, Ω) and therefore extend the result in [12] to also cover the case p = 1. We follow the strategy used in [12]. A crucial step in [12] is to prove that

1,p capp(E,F, Ω) = Nloc − capp(E,F, Ω),

1,p where in the latter case the test functions u are required to be in Nloc (Ω). We prove that the above equality also holds for p = 1. First we recall the following theorem. Theorem 3.1 ([32, Theorem 1.11]). Let X be a complete metric space that supports a doubling Borel measure µ which is nontrivial and ﬁnite on balls. Assume that X supports a weak (1, p)-Poincar´einequality for some 1 ≤ p < ∞. If an extended real valued function u : X → [−∞, ∞] has a p-integrable upper gradient, then u is measurable and locally p- integrable. The following observation is an easy consequence of the previous theorem.

Corollary 3.2. Let Ω ⊂ X be an open connected set. If u :Ω → [0, 1] has an upper gradient g ∈ Lp(Ω), then u is measurable. Proof. Recall that we have as a standing assumption that X supports a weak (1, 1)- Poincaréinequality. It suffices to prove that u coincides with a measurable function locally. To do so, for x0 ∈ Ω let r > 0 such that B(x0, 4r) ⊂ Ω, and let η be a 1/r-Lipschitz function on X such that 0 ≤ η ≤ 1, η = 1 on B(x0, r), and η = 0 on X \ B(x0, 2r). Consider the function v = ηu; note that while u is not defined on X \ Ω, since η = 0 on X \ Ω we can extend v by zero to X \ Ω. −1 A direct computation shows that ρ = ηg + r u χB(x0,2r)\B(x0,r) is an upper gradient of v in X. Thus it follows from Theorem 3.1 that v is measurable on X, and hence v is measurable on the ball B(x0, r) ⊂ Ω.; the proof is completed by noting that on B(x0, r) we have u = v. Theorem 3.3. Let E and F be nonempty disjoint compact subsets of an open connected set Ω ⊂ X. Then 1,1 cap1(E,F, Ω) = Nloc − cap1(E,F, Ω).

8 Proof. It is clear that

1,1 cap1(E,F, Ω) ≤ Nloc − cap1(E,F, Ω). Let ε > 0 and u :Ω → [0, 1] be a function such that u = 1 in E and u = 0 in F and with an upper gradient gu for which Z gudµ < cap1(E,F, Ω) + ε. Ω By Corollary 3.2, u is a measurable function with an integrable upper gradient. Since u 1,1 is bounded it follows that u ∈ Nloc (Ω). Hence Z 1,1 Nloc − cap1(E,F, Ω) ≤ gudµ < cap1(E,F, Ω) + ε. Ω The desired inequality follows by letting ε → 0. As pointed out in the beginning of Section 2, a complete metric measure space X with a doubling measure is proper, i.e. all closed and bounded subsets of X are compact. The doubling condition of the measure and weak (1, p)-Poincar´einequality, with 1 ≤ p < ∞, imply also that Lipschitz functions are dense in N 1,p(X), see for example [19, Theorem 4.1]. For the following theorem we refer to [29, Theorem 1.1]. Theorem 3.4. Let X be proper, Ω ⊂ X open, 1 ≤ p < ∞ and assume that continuous 1,p 1,p functions are dense in N (X). Then every u ∈ Nloc (Ω) is quasicontinuous in Ω; that is, for every ε > 0 there is an open set Uε with Capp(Uε) < ε such that u|Ω\Uε is continuous on Ω \ Uε.

Here, by the total capacity Capp(Uε) we mean the number

p Capp(Uε) = inf kukN 1,p(X),

1,p where the inﬁmum is taken over all u ∈ N (X) with u = 1 on Uε and 0 ≤ u ≤ 1. The following lemma (for 1 < p < ∞) is from [12]; we will prove that it also holds true in the case p = 1. A similar proof can be used to show that whenever E ⊂ X, we have Capp(E) = Capp,O(E), where the latter is obtained by considering test functions u as in the deﬁnition of Capp(E) but with the additional condition that u ≥ 1 in a neighborhood of E. See [33] and [29] for further discussion. Lemma 3.5. Let E and F be nonempty disjoint compact subsets of Ω. For every 0 < 1 ε < 2 there exists disjoint compact sets Eε and Fε of Ω such that for some δ > 0 S (i) x∈E B(x, δ) = Eε,

S (ii) x∈F B(x, δ) = Fε,

1,1 1 1,1 (iii) Nloc − cap1(Eε,Fε, Ω) ≤ 1−2ε Nloc − cap1(E,F, Ω) + 2ε . 9 1,1 Proof. Let 0 < ε < 1/2 and u ∈ Nloc (Ω) be such that 0 ≤ u ≤ 1, u = 1 in E, u = 0 in F , and with an upper gradient gu satisfying Z 1,1 gudµ < Nloc − cap1(E,F, Ω) + ε. Ω Let L = 4 dist(E,F )−1. By Theorem 3.4 the function u is quasicontinuous in Ω. Hence there exists an open set U such that u|Ω\U is continuous in Ω\U and Cap1(U) < ε/(1+L). Thus there exists v ∈ N 1,1(X), 0 ≤ v ≤ 1, such that v = 1 in U and Z Z ε v dµ + gvdµ < . X X 1 + L

Let E1 = E \ U and F1 = F \ U; then E1,F1 are compact subsets of Ω \ U with E \ E1 = E ∩ U and F \ F1 = F ∩ U subsets of U. Let

δ1 = dist (E1, {x ∈ Ω \ U : u(x) < 1 − ε})

δ2 = dist (F1, {x ∈ Ω \ U : u(x) > ε}) and let

δ = min {δ1, δ2, dist(E,F )/10, dist(E,X \ Ω)/2, dist(F,X \ Ω)/2} .

Since E1 and F1 are compact and u|Ω\U is continuous, it follows that δ > 0. Let [ Eε = B(x, δ) x∈E [ Fε = B(x, δ) x∈F and let η be an L-Lipschitz function such that −1 ≤ η ≤ 1, η = −1 in Fε, and η = 1 in Eε. Let u + ηv − ε w = max 0, e 1 − 2ε 1,1 and let w = min{1, we}. Now w ∈ Nloc (Ω) with w = 0 in Fε and w = 1 in Eε. Since w has an upper gradient g Lv + g g ≤ u + v , w 1 − 2ε 1 − 2ε we obtain Z 1,1 Nloc − cap1(Eε,Fε, Ω) ≤ gwdµ Ω 1 Z Z Z = gudµ + L v dµ + gvdµ 1 − 2ε Ω Ω Ω 1 2ε < N 1,1 − cap (E,F, Ω) + 1 − 2ε loc 1 1 − 2ε We recall the following result [29, Theorem 5.9]. The proof given there is valid for all 1 ≤ p < ∞. 10 Theorem 3.6 ([29, Theorem 5.9]). Let X be proper and assume that locally Lipschitz 1,p 1,p functions are dense in N (X). If Ω ⊂ X is open, u ∈ Nloc (Ω), and ε > 0, then there exists a locally Lipschitz function v :Ω → R such that ku − vkN 1,p(Ω) < ε. An analog of the next lemma for p > 1 was proved in [12].

Lemma 3.7. Let ε, δ, Eε, Fε, E, F and Ω be as in Lemma 3.5. Then

1,1 Nloc − cap1(Eε,Fε, Ω) ≥ locLip − cap1(E,F, Ω).

Proof. Let η1 be a Lipschitz function such that 0 ≤ η1 ≤ 1, η1 = 1 in E and η1 = 0 in Ω \ Eε. Correspondingly, let η2 be a Lipschitz function such that 0 ≤ η2 ≤ 1, η2 = 1 in 1,1 F and η2 = 0 in Ω \ Fε. Let γ > 0 and let u ∈ Nloc (Ω) be such that 0 ≤ u ≤ 1, u = 1 in Eε and u = 0 in Fε with an upper gradient gu such that Z 1,1 gudµ < Nloc − cap1(Eε,Fε, Ω) + γ. Ω

By Theorem 3.6 there is a sequence of locally Lipschitz functions ui, 0 ≤ ui ≤ 1 such that lim kui − ukN 1,1(Ω) = 0. i→∞ Let vi = (1 − η1 − η2)ui + η1 = (1 − ui)η1 + ui(1 − η2), for i = 1, 2,... Observe that vi = 1 in E, vi = 0 in F and vi has an upper gradient

gvi ≤ gη1 (1 − ui) + η1gui + gui (1 − η2) + uigη2

≤ gη1 (1 − ui) + η1(gui−u + gu) + (gui−u + gu)(1 − η2) + uigη2 .

Since we may assume that gu = 0 in Eε ∪ Fε, gη1 = 0 in Ω \ Eε, gη2 = 0 in Ω \ Fε and since u = 1 in Eε and u = 0 in Fε we have the following estimate Z

locLip − cap1(E,F, Ω) ≤ gvi dµ Ω Z Z ∞ ≤ kgη1 kL (Ω) |ui − u|dµ + gui−udµ Eε Eε Z Z Z

+ gudµ + gui−udµ + gudµ Eε Ω Ω Z ∞ + kgη2 kL (Ω) |ui − u|dµ. Fε

We may assume that gu = 0 in Eε because u = 1 in Eε. Thus by letting i → ∞ we obtain Z locLip − cap1(E,F, Ω) ≤ gudµ Ω 1,1 ≤ Nloc − cap1(Eε,Fε, Ω) + γ. The claim now follows by letting γ → 0. 11 Finally we obtain the main result of this section. Theorem 3.8. If E and F are nonempty disjoint compact subsets of Ω, then

Mod1(E,F, Ω) = cap1(E,F, Ω) = locLip − cap1(E,F, Ω).

Proof. It suﬃces to prove that

locLip − cap1(E,F, Ω) ≤ cap1(E,F, Ω).

Let Eε and Fε be as in Lemma 3.5. Then by Lemma 3.7 and Lemma 3.5,

1,1 locLip − cap1(E,F, Ω) ≤ Nloc − cap1(Eε,Fε, Ω) 1 ≤ N 1,1 − cap (E,F, Ω) + 2ε . 1 − 2ε loc 1 According to Theorem 3.3 we have that

1,1 Nloc − cap1(E,F, Ω) = cap1(E,F, Ω) and the claim now follows by letting ε → 0.

4. Variational BV-capacity

In this section we study different types of variational BV-capacities in metric measure spaces. Since the class BV(X) is less restrictive than N 1,1(X) in terms of pointwise behavior of functions, it is not obvious which definitions of BV-capacity are equivalent. Our main focus in this section is to study which pointwise conditions the test functions of capacity must satisfy in order to establish a BV-capacity that is comparable to the variational 1-capacity known in literature. For the Sobolev 1-capacity, due to the quasicontinuity of Sobolev type functions, one can either insist on the test functions u satisfying u = 1 in a neighborhood of the set whose capacity is being computed, or merely require that the test functions satisfies u = 1 on the set. Thus the two seemingly different notions of 1-capacity are equal; for instance, see the earlier discussion in Section 3. In the Euclidean setting Federer and Ziemer [3] studied a version of BV-capacity of a set E based on minimizing the perimeter of sets that contain E in the interior, and another version of BV-capacity based on test functions u that satisfy the requirement that the level set {u ≥ 1} should have positive upper density Hn−1-a.e. in E. They proved that these two quantities and the variational 1-capacity are equal. In [4] Kinnunen et.al. studied the following variational BV-capacity in a metric measure space setting. Definition 4.1. Let K ⊂ X be compact. The variational BV-capacity of K is defined as capBV(K) = inf kDuk(X), where the infimum is taken over all u ∈ BV(X) such that u = 1 on a neighborhood of K, 0 ≤ u ≤ 1, and the support of u is a compact subset of X.

12 It was shown in [4] that when K is compact the above BV-capacity is comparable to the variational 1-capacity and to the Hausdorff content of codimension one. In this section, we define two modifications of BV-capacity. First we recall the definitions of the approximate upper and lower limits. Let u be a measurable function. Define ∨ µ (B(x, r) ∩ {u > t}) u (x) = inf t ∈ R : lim = 0 , r→0 µ (B(x, r)) and ∧ µ (B(x, r) ∩ {u < t}) u (x) = sup t ∈ R : lim = 0 . r→0 µ (B(x, r)) Moreover, we define u∨ + u∧ u = . 2 If u∨(x) = u∧(x), we denote by

ap lim u(y) = u∨(x) y→x the approximate limit of u at point x. The function u is approximately continuous at x if ap lim u(y) = u(x). y→x The jump set of function u, in the sense of approximate limits, is deﬁned as

∨ ∧ Su = {x ∈ X : u (x) 6= u (x)} .

n The classical result states that Su is countably (n − 1)-rectifiable for u ∈ BV(R ), see for instance [10, Theorem 5.9.6]. This result has an analog in the metric setting, where S the countable (n − 1)-rectifiability is replaced with Su = S(k) with H(S(k)) < ∞ k∈N for each k ∈ N; this follows from [18, Proposition 5.2 and Theorem 4.4]. We consider the following capacities in this section. Definition 4.2. Let Ω ⊂ X be an open set and K ⊂ Ω be a compact set. We define Z cap1,O(K, Ω) = inf gudµ Ω

1,1 where the infimum is taken over all weak upper gradients gu of functions u ∈ N (Ω) such that supt(u) is a compact subset of Ω with u = 1 on a neighborhood of K. Moreover, we define capBV,O(K, Ω) = inf kDuk(Ω) where the infimum is taken over all u ∈ BV(Ω) such that 0 ≤ u ≤ 1, supt(u) is a compact subset of Ω with u = 1 on a neighborhood of K. In this definition, as with cap1,O(·, Ω), we can remove the condition that 0 ≤ u ≤ 1. We define

capgBV(K, Ω) = inf kDuk(Ω)

13 where the inﬁmum is taken over all u ∈ BV(Ω) such that supt(u) is a compact subset of Ω and u ≥ 1 on K. Finally we deﬁne

capgBV−trunc(K, Ω) = inf kDuk(Ω) where the inﬁmum is taken over all u ∈ BV(Ω) such that 0 ≤ u ≤ 1, supt(u) is a compact subset of Ω and u ≥ 1 on K. Remark. Note that we do not assume that these functions satisfy the condition u ≤ 1 in the deﬁnition of capgBV(K, Ω), though we can assume that u ≥ 0. For any compact set K we automatically have the following inequalities:

capgBV(K, Ω) ≤ capgBV−trunc(K, Ω) ≤ capBV,O(K, Ω) ≤ cap1,O(K, Ω). The ﬁrst inequality immediately follows from comparing the test functions. The second inequality follows from the observation that if u = 1 in some open neighborhood of a compact set K, then u∧(x) ≥ 1 for every x ∈ K. The third inequality follows from the fact that N 1,1(Ω) ⊂ BV(Ω). Theorem 4.3. For any compact set K ⊂ Ω we have

cap1,O(K, Ω) = capBV,O(K, Ω).

Proof. As pointed out above, capBV,O(K, Ω) ≤ cap1,O(K, Ω). In order to prove the opposite inequality, let ε > 0 and u ∈ BV(Ω) be such that u = 1 in an open neighborhood U b Ω of K, supt(u) is a compact subset of Ω, and that

kDuk(Ω) < capBV,O(K, Ω) + ε. 1 According to Lemma 2.6, there exist functions ui ∈ Lip(Ω) such that ui → u in L (Ω) and satisfying Z

lim gui dµ = kDuk(Ω). i→∞ Ω We may assume that 0 ≤ ui ≤ 1 and by Lemma 2.6, the functions ui all have a compact 0 0 support in Ω. Let U be an open set such that K ⊂ U b U and η be a Lipschitz function such that 0 ≤ η ≤ 1, η = 1 in U 0 and η = 0 in Ω \ U. We deﬁne

vi = η + (1 − η)ui = (1 − ui)η + ui 1,1 for i = 1, 2,... and note that vi ∈ N (Ω) with compact support in Ω. Furthermore, the absolute continuity of vi on p-modulus almost every path yields that for every i ∈ N the function vi has an upper gradient gvi satisfying

gvi ≤ |(1 − ui)gη + (1 − η)gui | = (1 − ui)gη + (1 − η)gui , see [34, Lemma 3.1] and [22, Theorem 2.11]. Since we may assume that gη = 0 in Ω \ U we have the following estimate Z Z Z

lim sup gvi dµ ≤ lim sup (1 − ui)gηdµ + lim sup (1 − η)gui dµ i→∞ Ω i→∞ Ω i→∞ Ω Z ≤ kgηk∞ lim sup |u − ui|dµ + kDuk(Ω) i→∞ U

= kDuk(Ω) < capBV,O(K, Ω) + ε. 14 Thus, for every ε > 0 we ﬁnd an admissible N 1,1(Ω)–function with a compact support in Ω such that Z

cap1,O(K, Ω) ≤ lim sup gvi dµ ≤ kDuk(Ω) < capBV,O(K, Ω) + ε. i→∞ Ω The claim now follows by letting ε → 0. Before proceeding any further in the study of comparisons of capacities, we give a counterexample demonstrating that capgBV(K, Ω) 6= capBV,O(K, Ω) in general. In this example we use the fact that if a ball B ⊂ E, where E ⊂ R2 is a bounded set with ﬁnite perimeter, then P (B) ≤ P (E). This type of results for convex sets are part of the general folklore. For the details of one such result which is suﬃcient for our purposes, see [35, Proof of Proposition 3.5 and Appendix]. Example 4.4. Let m denote the ordinary Lebesgue measure and let X = (R2, | · |, µ) where dµ = w dm with ( 1 when |x| ≤ 1, w(x) = 8 when |x| > 1.

Let K = B(0, 1) and Ω = B(0, 10). Let ε > 0. The coarea formula implies that

capBV,O(K, Ω) ≥ kDuk(Ω) − ε ≥ P ({u > t}, Ω) − ε for some admissible function u ∈ BV(Ω) and for some 0 < t < 1. Furthermore, since u is compactly supported in Ω, via zero extension we obtain that

capBV,O(K, Ω) ≥ P ({u > t},X) − ε.

From the fact that u = 1 in a neighborhood of B(0, 1) it follows that {u > t} c B(0, 1+δ) for some δ > 0. Since the perimeter measure vanishes outside the measure theoretic boundary, we can apply the result mentioned in the discussion preceding this example also in this weighted case for sets {u > t} and B(0, 1 + δ) to obtain P ({u > t},X) − ε ≥ P (B(0, 1 + δ),X) − ε = 16π + 16πδ − ε. Thus letting ε → 0, we get

capBV,O(K, Ω) ≥ 16π.

By approximating 2χK with Lipschitz functions

ui = 2 min {1, max {0, −2i|x| + 2i − 1}} , see [6, Example 4.5] for details, it follows that Z capgBV(K, Ω) ≤ kD(2χK )k(Ω) ≤ lim inf |Dui|dµ ≤ 4π. i→∞ Ω Therefore, in this example we get

capgBV(K, Ω) < capBV,O(K, Ω). 15 In the light of the above example, our goal now is to show that these capacities are comparable. To do so, by the previous observations we need to prove that there is a constant C > 0 such that 1 cap (K, Ω) ≤ cap (K, Ω) ≤ C cap (K, Ω) C BV,O gBV−trunc gBV for every compact set K ⊂ Ω. We start with the following theorem. In the proof we combine discrete convolution and boxing inequality type arguments to obtain the desired result. Theorem 4.5. There exists a constant C > 0, depending only on the constants in the (1, 1)-Poincar´einequality and the doubling constant of the measure, such that

capBV,O(K, Ω) ≤ C capgBV−trunc(K, Ω) for any compact set K ⊂ Ω. Proof. Let u ∈ BV(Ω), with 0 ≤ u ≤ 1, be an admissible function for computing capgBV−trunc(K, Ω), such that

kDuk(Ω) < capgBV−trunc(K, Ω) + ε. Since u ≥ 1 on K and 0 ≤ u ≤ 1, we have u∧ ≥ 1 on K. So by the deﬁnition of u∧, for every x ∈ K and t < 1 we have µ (B (x, r) ∩ {u < t}) lim = 0. r→0 µ (B(x, r))

For t ≥ 0 set Et = {u > t}. Thus for every x ∈ K and 0 ≤ t < 1, µ (B (x, r) ∩ E ) lim t = 1. (2) r→0 µ (B(x, r))

By the coarea formula there exists 0 < t0 < 1 such that

P (E , Ω) ≤ kDuk(Ω) < cap (K, Ω) + ε. (3) t0 gBV−trunc Let µ (B (x, r) ∩ E ) E = x ∈ Ω : lim t0 = 1 . r→0 µ (B(x, r)) Observe that dist (E,X \ Ω) = δ > 0, and due to the Lebesgue diﬀerentiation theorem,

µ(E∆Et0 ) = 0. Thus P (Et0 , Ω) = P (E, Ω) = P (E,X). The equation (2) yields that K ⊂ E. We now apply discrete convolution, as in the proof of Theorem 6.4 in [4], to the function w = χE with parameter ρ = δ/(40τ), where τ is the constant in the weak (1, 1)-Poincar´einequality, to obtain

∞ X v(x) = ϕi(x)wBi . i=1

∞ Here {Bi}i=1 is the covering related to the discrete convolution. 16 Let us make few comments on this technique. The radius of each ball Bi is ρ = ∞ δ/(40τ). The partition of unity {ϕi}i=1 can be constructed in such way that ϕi is a C/δ-Lipschitz function with ϕi ≥ 1/C in Bi and supt(ϕi) ⊂ 2Bi for every i = 1, 2,..., where the constant C depends only on the overlap constant of the covering related to the discrete convolution, see [4]. The overlap constant does not depend on the radii of the balls used in the covering. We also point out that in [4] the authors assume that µ(X) = ∞, but the construction and properties of discrete convolution do not depend on this assumption at all, and hence that assumption is not needed in this paper. The proof of Theorem 6.4 in [4] implies that v ∈ N 1,1(Ω) ∩ C(Ω) with 0 ≤ v ≤ 1 and supt(v) is a compact subset of Ω. We divide the set E into two parts µ (Bi ∩ E) 1 E1 = x ∈ E : > for some Bi with x ∈ Bi µ(Bi) 2 and µ (Bi ∩ E) 1 E2 = x ∈ E : ≤ for every Bi with x ∈ Bi . µ(Bi) 2

Let x ∈ E1. By the deﬁnition of E1 there is a ball Bi such that x ∈ Bi and wBi > 1/2. Since ϕi ≥ 1/C in Bi we have that

v ≥ ϕiwBi ≥ 1/(2C) > 0 in Bi and therefore ve = 2Cv ≥ 1 in a neighborhood of the set E1. Note that ve ∈ N 1,1(Ω) ⊂ BV(Ω) with compact support in Ω and with the total variation

kDvek(Ω) ≤ CkDχEk(Ω) = CP (E, Ω), see the proof of [4, Theorem 6.4] for details. Let x ∈ E2 and Bi be such that x ∈ Bi. Let B = B(x, ρ) and notice that Bi ⊂ 2B ⊂ 4Bi. Thus we can estimate µ (2B ∩ E) µ (2B) µ (2B \ E) = − µ (2B) µ (2B) µ (2B) µ (B \ E) ≤ 1 − i µ (4Bi)

µ (Bi \ E) ≤ 1 − 2 . CDµ (Bi)

Here CD is the doubling constant of the measure µ. By the deﬁnition of E2 we know that µ(B \ E) 1 i ≥ . µ(Bi) 2 Hence µ (2B ∩ E) 1 ≤ 1 − 2 = c < 1. µ (2B) 2 CD

Since x ∈ E2 is a point of Lebesgue density 1 for Et0 and µ(E∆Et0 ) = 0, we have that µ (B (x, r) ∩ E) lim = 1. r→0 µ (B(x, r)) 17 We now follow the proof of the boxing inequality [4, Theorem 3.1], see also [36, Lemma 3.1]. For every x ∈ E2 there exists rx with 0 < rx ≤ 2ρ such that µ (B (x, r ) ∩ E) x ≤ c µ (B(x, rx)) and µ (B (x, r /2) ∩ E) x > c. µ (B(x, rx/2))

From the choice of rx it follows that

µ (B(x, rx) \ E) = µ (B(x, rx)) − µ (B(x, rx) ∩ E)

≥ µ (B(x, rx)) − c µ (B(x, rx))

= (1 − c)µ (B(x, rx)) .

Furthermore, we have that

µ (B (x, r ) ∩ E) µ (B (x, r /2) ∩ E) c x ≥ x > . µ (B(x, rx)) CDµ (B(x, rx/2)) CD Thus by the relative isoperimetric inequality we obtain

µ (B(x, r )) min {µ (B (x, r ) ∩ E) , µ (B(x, r ) \ E)} x ≤ C x x rx rx ≤ CP (E,B (x, τrx)) , where the constant C depends only on the doubling constant of the measure and the constants in the weak (1, 1)-Poincar´einequality. We apply the standard covering argument to the family of balls B(x, τrx), x ∈ E2, to obtain pairwise disjoint balls B(xj, τrj), j ∈ N, such that ∞ [ [ E2 ⊂ B(x, τrx) ⊂ B(xj, 5τrj).

x∈E2 j=1 By using the doubling property of µ and the fact that P (E, ·) is a Borel measure and that the balls B(xj, τrj) are pairwise disjoint, we obtain

∞ ∞ X µ (B(xj, 5τrj)) X µ (B(xj, rj)) ≤ C 5τr r j=1 j j=1 j ∞ X ≤ C P (E,B (xj, τrj)) j=1

 ∞  [ = CP E, B(xj, τrj) j=1 ≤ CP (E, Ω).

18 For positive integers j we deﬁne functions ψj ∈ Lip(Ω) by dist (x, B(xj, 5τrj)) ψj(x) = 1 − . 5τrj + Let ψ = sup ψj 1≤j<∞ and observe that ψ = 1 in a neighborhood of the set E2. The radius of the covering balls B(xj, 5τrj) satisﬁes 5τrj ≤ δ/4 and since δ = dist(E,X \ Ω) ≤ dist(E2,X \ Ω), we know that ψ has compact support in Ω. Moreover, ψ has an upper gradient 1 gψ = sup χB(xj ,10τrj ), 1≤j<∞ 5τrj see [22, Lemma 1.28]. For the upper gradient we have the estimate

∞ Z Z X 1 g dµ ≤ χ dµ ψ 5τr B(xj ,10τrj ) Ω Ω j=1 j ∞ X Z 1 = dµ 5τr j=1 B(xj ,10τrj ) j ∞ X µ(B(xj, 5τrj)) ≤ C D 5τr j=1 j ≤ CP (E, Ω).

Thus ψ ∈ N 1,1(Ω). Let ν = min{ve + ψ, 1} where ve was constructed in considering E1, and notice that ν ∈ N 1,1(Ω) ⊂ BV(Ω), 0 ≤ ν ≤ 1 and supt(ν) is a compact subset of Ω. Moreover ν = 1 in a neighborhood of E ⊃ K. By using the properties of total variation measure we have that

cap1,O(K, Ω) ≤ kDνk(Ω) ≤ kDvek(Ω) + kDψk(Ω) Z ≤ kDvek(Ω) + gψdµ Ω ≤ CP (E, Ω).

In the previous estimate the third inequality follows from the fact that kDψk(Ω) ≤ 1,1 1 kgψkL (Ω) for ψ ∈ N (Ω), see [30, Theorem 6.2.2]. Since P (E, Ω) = P (Et0 , Ω), the inequality (3) yields cap1,O(K, Ω) = capBV,O(K, Ω) < C capgBV−trunc(K, Ω) + ε . The claim now follows by letting ε → 0.

Thus capBV,O(·, Ω) and capgBV−trunc(·, Ω) are comparable with a constant which does not depend on Ω. We obtain the remaining inequality in the next theorem. 19 Theorem 4.6. There exists a constant C > 0, depending only on the constants in the (1, 1)-Poincar´einequality and the doubling constant of the measure, such that

capgBV−trunc(K, Ω) ≤ C capgBV(K, Ω) for every compact set K ⊂ Ω.

Proof. Let ε > 0 and u ∈ BV(Ω) such that supt(u) is a compact subset of Ω, u ≥ 1 in K, and kDuk(Ω) < capgBV(K, Ω) + ε. We may assume that 0 ≤ u ≤ 2 since the truncated function ue = min{u, 2} is an admissible function with smaller total variation. This is due to the observations that for ∨ any t < 2 we have {u > t} = {ue > t} and {u < t} = {ue < t}. Hence u (x) ≥ 2 implies ∨ ∨ ∨ ∨ ∧ ∧ that ue (x) = 2, and if u (x) < 2 then u (x) = ue (x) and u (x) = ue (x). We divide the set K into two parts;

∨ ∧ K1 = {x ∈ K : u (x) − u (x) < 1/2} and ∨ ∧ K2 = {x ∈ K : u (x) − u (x) ≥ 1/2} .

Note that K2 ⊂ Su and thus by [18, Theorem 5.3], Z 2 Z H(K2) ≤ dH ≤ θudH K2 α K2

≤ CkDuk(Su) ≤ CkDuk(Ω). (4)

Here C = 2/α, and we used the estimate that for x ∈ K2

u∨(x) Z α θ (x) ≥ α dt ≥ , u 2 u∧(x) where θu is defined as in [18, Theorem 5.3] and α > 0 depends only on the doubling constant of the measure and the constants related to the weak (1, 1)-Poincaréinequality, see [18, Theorem 4.4] and [18, Theorem 5.3]. Hence the constant C in the above estimate depends only on the doubling constant of the measure and the constants related to the Poincaréinequality. Thus for δ > 0 there is a covering

∞ [ B(xi, ri) ⊃ K2 i=1 such that ri ≤ δ for every i = 1, 2,... with the estimate

∞ X µ (B (xi, ri)) < H(K ) + ε. (5) r 2 i=1 i 20 By choosing δ to be small, we may assume that

dist (K,X \ Ω) δ < . 10

We construct an admissible test function to estimate capgBV−trunc(K, Ω) as follows. Let us deﬁne functions ϕi by dist (x, B (xi, ri)) ϕi(x) = 1 − ri + for i = 1, 2,... and ϕ0 = min {2u, 1} . Let ϕ = sup ϕi 0≤i<∞ and observe that 0 ≤ ϕ ≤ 1 and supt(ϕ) is a compact subset of Ω. If x ∈ K2, then x ∈ B(xi, ri) for some index 1 ≤ i < ∞ and ϕ ≥ ϕi ≥ 1 in this neighborhood of x. Thus ∨ ∧ ϕ (x) ≥ ϕ (x) ≥ 1 and so ϕ(x) ≥ 1. To obtain similar estimate for the points of set K1 notice that for t < 1 we have

{ϕ < t} ⊂ {ϕ0 < t} ⊂ {2u < t} .

∧ Since u ≥ 1/2 in K1, it follows that for every x ∈ K1 and for any t < 1/2,

µ (B(x, r) ∩ {u < t}) lim = 0. r→0 µ (B(x, r))

Thus for every x ∈ K1 and for any t < 1 we have that µ (B(x, r) ∩ {2u < t}) lim = 0, r→0 µ (B(x, r)) and hence µ (B(x, r) ∩ {ϕ < t}) lim = 0. r→0 µ (B(x, r)) ∧ This implies that ϕ ≥ 1 in K1 and therefore ϕ ≥ 1 in K1. We have thus obtained that ϕ ≥ 1 in K. Let ψk = max ϕi 0≤i≤k and note that ψk → ϕ pointwise as k → ∞. We apply the dominated convergence theorem to obtain Z Z lim |ϕ − ψk|dµ = lim (ϕ − ψk) dµ = 0 k→∞ Ω Ω k→∞

21 1 and thus ψk → ϕ in L (Ω) as k → ∞. By the properties of the total variation measure we have

kDϕk(Ω) ≤ lim inf kDψkk(Ω) k→∞ k X ≤ lim inf kDϕik(Ω) k→∞ i=0 ∞ X = kDϕ0k(Ω) + kDϕik(Ω). i=1

Using the deﬁnitions of functions ϕi we have the estimates

kDϕ0k(Ω) ≤ 2kDuk(Ω), and by (5) and (4),

∞ ∞ X X Z 1 kDϕ k(Ω) ≤ dµ i r i=1 i=1 B(xi,2ri) i ∞ X µ (B (xi, ri)) ≤ C D r i=1 i

≤ CD (CkDuk(Ω) + ε) ≤ C (kDuk(Ω) + ε) .

Thus

capgBV−trunc(K, Ω) ≤ kDϕk(Ω) ≤ C (kDuk(Ω) + ε)

≤ C (capgBV(K, Ω) + ε) . The claim now follows by letting ε → 0. The constant C in the previous theorem does not depend on Ω. We have thus obtained that there exists a constant C > 0 such that 1 cap (K, Ω) ≤ cap (K, Ω) ≤ C cap (K, Ω) C BV,O gBV−trunc gBV for any compact set K ⊂ Ω. Combining these estimates we may state the following. Corollary 4.7. Let Ω ⊂ X be open and K a compact subset of Ω. Then

cap1,O(K, Ω) = capBV,O(K, Ω) ≈ capgBV−trunc(K, Ω) ≈ capgBV(K, Ω), with the constants of comparison depending only on the doubling constant of the measure and the constants in the Poincar´einequality.

Thus, on the collection of all compact subsets of Ω we have cap1,O(·, Ω), capgBV(·, Ω), capgBV−trunc(·, Ω) and capBV,O(·, Ω) are equivalent by two sided estimates. 22 5. Codimension one Hausdorﬀ measure and Capacity

In this section we study comparisons between the variational capacity and the capacity where the norm of the function is also included. In the previous sections we studied relative capacities of compact sets. Should the metric space X be parabolic, then when Ω = X the relative capacities of every set are zero; see for example the discussion in [4, Section 7]. However, the total capacities considered in this section do not have this drawback. Furthermore, since capacities in general are not inner measures, it would be insuﬃcient for potential theory to consider only compact sets; in this section we consider more general sets. Let B ⊂ X be a ball. For E ⊂ B, we deﬁne the following four versions of capacity: Z cap1(E,B) = inf gu dµ, B

capBV,O(E,B) = inf kDuk(B), Z Cap1(E,B) = inf (|u| + gu) dµ, B Z CapBV,O(E,B) = inf |u| dµ + kDuk(B) , B where the first and third infimum are taken over all u ∈ N 1,1(B) and all upper gradients gu of u such that u = 1 on E and u has compact support in B, and the second and fourth infimum are taken over all u ∈ BV(B) such that u = 1 in a neighborhood of E and u has compact support in B. In each of the above definitions we consider the infimum to be infinite, if there are no such functions. These definitions are consistent with the notions studied in previous sections for the case that E is compact. Observe that when E b B we could also require the test functions to satisfy the condition u = 1 in a neighborhood of E in the definition of cap1(E,B) and Cap1(E,B) to obtain the same quantities. This fact can be proved by using a cutoff function and quasicontinuity of functions in N 1,1(B), see for instance [22, Theorem 6.19] for the proof of an analogous result. Remark. We could actually extend the above definitions of capacities to arbitrary sets E ⊂ X by requiring that the test functions satisfy the pointwise conditions in E ∩ B. Our goal is to prove that the above capacities have the same null sets. In order to compare variational capacities with the total capacities where the norm of the function is also included, we apply a metric measure space version of Sobolev inequality, see [37], [38, Lemma 2.10], [39, Proposition 3.1] and [22, Section 5.4] for general discussion. Lemma 5.1. Let B ⊂ X be a ball such that X \ B is nonempty. For every E ⊂ B we have cap1(E,B) ≈ Cap1(E,B) with the constants of comparison depending solely on the doubling constant, the Poincaré constants, and the ball B.

Proof. To see this, note that clearly cap1(E,B) ≤ Cap1(E,B). In order to prove the 1,1 remaining comparison, we may assume that cap1(E,B) < ∞. Suppose u is in N (B) with u = 1 on E and u has a compact support in B. Since X \B is nonempty, it contains 23 a ball which has a positive measure according to the assumptions we made at beginning of this article. Thus X \ B has a positive capacity and we can apply the “Poincaré 1,1 inequality for N0 (B)-functions”, see [22, Corollary 5.48], to obtain Z Z |u| dµ ≤ C gu dµ. B B Here the constant C depends on the ball B as well as the doubling constant of measure and constants in the weak (1, 1)-Poincaréinequality. In the above estimate we used the fact that since u has a compact support in B we can take gu = 0 on X \ B. Hence Z Z Cap1(E,B) ≤ (|u| + gu) dµ ≤ C gu dµ. B B Taking the infimum over all such u gives

Cap1(E,B) ≤ C cap1(E,B).

In fact, since kDuk(B) is defined using approximation by Lipschitz functions, we can use same arguments as in the above proof together with Lemma 2.6 to obtain an analogous result for the corresponding BV-capacities. In this case we have the inequality Z |u| dµ ≤ CkDuk(B) B and this yields CapBV,O(E,B) ≤ CkDuk(B). Thus we have the following. Corollary 5.2. Let B ⊂ X be a ball such that X \ B is nonempty. For every E ⊂ B we have capBV,O(E,B) ≈ CapBV,O(E,B), with the constants of comparison depending solely on the doubling constant, the Poincaré constants, and the ball B. Furthermore, if K ⊂ B is compact, then according to Theorem 4.3 in the previous section capBV,O(K,B) = cap1,O(K,B) = cap1(K,B). For more general sets E, we consider Z Cap1(E) = inf (|u| + gu) dµ, X Z CapBV,O(E) = inf u dµ + kDuk(X) X where the first infimum is taken over all u ∈ N 1,1(X) such that u = 1 on E and the second infimum is taken over all u ∈ BV(X) such that u = 1 on an open neighborhood E. It was shown in [4] that the set of non-Lebesgue points of a function in N 1,1(X) is of zero Cap1–capacity. 24 It can be seen that Cap1(E) = 0 if and only if for all balls B in X, Cap1(E ∩B, 2B) = 0. Indeed, let us assume that the latter condition holds. By comparing the test functions we immediately have that

Cap1(E ∩ B) ≤ Cap1(E ∩ B, 2B) for all balls B ⊂ X. Thus, if a ball B ⊂ X is ﬁxed, then the countable subadditivity of 1-capacity, see for instance [22], implies that

∞ ∞ X X Cap1(E) ≤ Cap1(E ∩ kB) ≤ Cap1(E ∩ kB, 2kB) = 0. k=1 k=1

In order to prove the opposite inequality, let us assume that Cap1(E) = 0. Let B ⊂ X be a ball and let ε > 0. There is a function u ∈ N 1,1(X) such that u = 1 on E and that Z (|u| + gu) dµ < ε. X Let v = uη where η ∈ Lip(2B) is such that 0 ≤ η ≤ 1, η = 1 in B and that supt(η) is a compact subset of 2B. Denote by L > 0 the Lipschitz constant of η. Then ηgu + L|u| is a weak upper gradient of v and Z Cap1(E ∩ B, 2B) ≤ (|v| + ηgu + L|u|) dµ 2B Z ≤ (L + 1) (|u| + gu) dµ X < (L + 1)ε.

The desired result then follows by letting ε → 0. The previous argument can be used to obtain the corresponding result for BV- capacities. Indeed the BV-capacity CapBV,O(·) is countable subadditive, see [6, The- orem 3.3], and thus CapBV,O(E ∩ B, 2B) = 0 for all balls B in X implies similarly that CapBV,O(E) = 0. On the other hand, if CapBV,O(E) = 0, then the equivalence of capacities, see [6, Theorem 4.3], implies that Cap1(E) = 0. Thus the previous statement concerning 1-capacities together with the fact that N 1,1(2B) ⊂ BV(2B) gives

CapBV,O(E ∩ B, 2B) ≤ Cap1(E ∩ B, 2B) = 0 for all balls B ⊂ X. Let us recall that the Theorems 4.3 and 5.1 in [6] imply that for any set E ⊂ X

CapBV,O(E) = 0 ⇐⇒ Cap1(E) = 0 ⇐⇒ H(E) = 0. The following corollary is a consequence of the results studied above in this section. In this result we assume that X \2B is nonempty. However, if X \2B is empty, then 2B = X and the following versions of total 1-capacities coincide, as do the total BV-capacities; however, the variational capacities are all zero. Corollary 5.3. Let B ⊂ X be a ball such that X \2B is nonempty and let E ⊂ B. Then the following are equivalent. 25 1. Cap1(E) = 0,

2. CapBV,O(E) = 0,

3. Cap1(E, 2B) = 0,

4. CapBV,O(E, 2B) = 0,

5. cap1(E, 2B) = 0,

6. capBV,O(E, 2B) = 0, 7. H(E) = 0.

In potential theory one is usually concerned with whether a set is of zero capacity or not. The above corollary shows that any of the six forms of capacity can be used in studying this question. Furthermore, if CapBV,O(E,B) = 0 then H(E) = 0. The above type total capacities and their connections to Hausdorﬀ measure of codimension one were studied in [6]. The relative variational BV-capacity in metric measure space setting was studied in [7] in the context of De Giorgi measure and an obstacle problem. In [7], the authors require the capacity test functions for capBV,O(E,B) to be zero in X \B. As an interesting result [7, Corollary 6.4] they obtain that if capBV,O(E,B) > 0, then ∗ H(∂ G) ≈ capBV,O(E,B), for some G ∈ G satisfying E ⊂ int G. The class G, related to De Giorgi measure, is a reﬁned collection of measurable sets satisfying certain density conditions, see [7] for details. Acknowldegement: N.S. was partially supported by the Taft Foundation of the University of Cincinnati. H.H. was supported by The Finnish National Graduate School in Mathematics and its Applications. Part of the research was conducted during H.H.’s visit to the University of Cincinnati and N.S.’s visit to Aalto University; the authors thank these institutions for their kind hospitality.

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